Nonconforming virtual element method for 2 m th order partial differential equations in $${\mathbb {R}}^n$$
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Nonconforming virtual element method for 2mth order partial differential equations in ℝn with m > n Xuehai Huang1 Received: 1 September 2019 / Revised: 15 September 2020 / Accepted: 19 September 2020 / Published online: 17 November 2020 © Istituto di Informatica e Telematica (IIT) 2020
Abstract The H m-nonconforming virtual elements of any order k on any shape of polytope in ℝn with constraints m > n and k ≥ m are constructed in a universal way. A generalized Green’s identity for H m inner product with m > n is derived, which is essential to devise the H m-nonconforming virtual elements. By means of the local H m projection and a stabilization term using only the boundary degrees of freedom, the H m-nonconforming virtual element methods are proposed to approximate solutions of the m-harmonic equation. The norm equivalence of the stabilization on the kernel of the local H m projection is proved by using the bubble function technique, the Poincaré inquality and the trace inequality, which implies the well-posedness of the virtual element methods. The optimal error estimates for the H m-nonconforming virtual element methods are achieved from an estimate of the weak continuity and the error estimate of the canonical interpolation. Finally, the implementation of the nonconforming virtual element method is discussed. Keywords H m-nonconforming virtual element · Generalized Green’s identity · Polyharmonic equation · Norm equivalence · Error analysis Mathematics Subject Classification 65N30 · 65N12 · 65N22
1 Introduction The H m-nonconforming virtual elements of order k ∈ ℕ on a very general polytope K ⊂ ℝn in any dimension and any order with constraints m ≤ n and k ≥ m have been devised in [17]. While an important case m = 3 and n = 2 , i.e. the triharmonic equation in two dimensions is not involved in [17]. To this end, and also for theoretical considerations, we will study the H m-nonconforming virtual element (K, NK , VK ) for case m > n in this paper, which can be considered as the second part of the work [17]. Here * Xuehai Huang [email protected] 1
School of Mathematics, Shanghai University of Finance and Economics, Shanghai 200433, China
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NK is the set of degrees of freedom, and VK the finite-dimensional space of shape functions. The virtual element can be defined on polytopes of any shape, and thus allows the division of the domain into different type of polytopes [9, 10], which makes the discrete method easier to capture the singularity of the solution. The key feature of the virtual element method is that it is completely determined by the degrees of freedom, and the virtual element space is only used for the analysis rather than entering the discrete method for elliptic problems. It is arduous to design H m-conforming or nonconforming finite elements for large k and m, especially m > n . With the help of the bubble functions, Wu and Xu constructed the minimal H m-nonconforming finite elements on simplices in any dimension with m = n + 1 in [31].
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